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Question: If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)$, prove that $\frac{d y}{d x}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}$. Solution: We are given with an equation $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)$, we have to prove that $\frac{d y}{d x}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}$ by using the given equation we will first find the value of $\frac{d y}{d x}$ and we will put this in the equation we have to prove, so by differentiating the equation on both sides with respect to $x$, we get, Put $x=\sin A$ and $y=\s...
Read More →Arrange the following elements
Question: Arrange the following elements in the order of their increasing non-metallic character Li, O, C, Be, F (a)F O C Be Li (b)Li Be C O F (c)F O C Be Li (d)F O Be C Li. Solution: (b).The non-metallic character of the elements increases along a period. These elements are present in second period in the order Li, Be, C, O, F....
Read More →Hari bought 20 kg of rice at ₹ 36 per kg and 25 kg of rice at ₹ 32 per kg.
Question: Hari bought 20 kg of rice at ₹ 36 per kg and 25 kg of rice at ₹ 32 per kg. He mixed the two varieties and sold the mixture at ₹ 38 per kg. Find his gain per cent in the whole transaction. Solution: Total cost of rice of 1st variety = ₹36/kg 20 kg = ₹720 Total cost of rice of 2nd variety = ₹32/kg 25 kg = ₹800 Total cost of the two rice varieties = ₹720 + ₹800 = ₹1,520 Total quantity of the two rice varieties = 20 kg + 25 kg = 45 kg Selling price of the mixture of two rice = ₹38/kg 45 kg...
Read More →Arrange the following elements
Question: Arrange the following elements in the order of their decreasing metallic character Na, Si, Cl, Mg, Al (a)Cl Si Al Mg Na (b)Na Mg Al Si Cl (c)Na Al Mg Cl Si (d)Al Na Si Ca Mg. Solution: (b).The metallic character of the elements decreases along a period. These elements are present in third period...
Read More →Find the domain and the range of each of the following real
Question: Find the domain and the range of each of the following real function: $f(x)=\frac{3 x-2}{x+2}$ Solution: Given: $f(x)=\frac{3 x-2}{x+2}$ Need to find: Where the functions are defined. Let, $f(x)=\frac{3 x-2}{x+2}=y$ .......(1) To find the domain of the function f(x) we need to equate the denominator of the function to 0. Therefore $x+2=0$ $\Rightarrow x=-2$ It means that the denominator is zero when $x=-2$ So, the domain of the function is the set of all the real numbers except $-2$. T...
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Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ in each of the following: $\sin x y+\cos (x+y)=1$ Solution: We are given with an equation $\sin x y+\cos (x+y)=1$, we have to find $\frac{d y}{d x}$ of it, so by differentiating the equation on both sides with respect to $x$, we get, $\cos x y\left(y+x \frac{d y}{d x}\right)-\sin (x+y)\left(1+\frac{d y}{d x}\right)=0$ $\frac{d y}{d x}[x \cos x y-\sin (x+y)]=\sin (x+y)-y \cos x y$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\sin (\mathrm{x}+\mathrm{y})-...
Read More →Robin purchased an old car for Rs 73500.
Question: Robin purchased an old car for Rs 73500. He spent Rs 10300 on repairs and paid Rs 2600 for its insurance. Then he sold it to a mechanic for Rs 84240. What was his percentage gain or loss? Solution: CP of the car $=$ Rs. 73500 Repairs $=$ Rs. 10300 Insurance $=$ Rs. 2600 Total CP $=73500+10300+2600=$ Rs. 86400 SP $=$ Rs. 84240 Since SP $$ CP, Robin has a loss. Loss $=86400-84240$ $=$ Rs. 2160 Loss percentage $=\left(\frac{\text { loss }}{\text { total CP }} \times 100\right) \%$ $=\left...
Read More →Which of the following are the characteristics
Question: Which of the following are the characteristics of isotopes of an element ? (i) Isotopes of an element have same atomic mass. (ii) Isotopes of an element have same atomic number. (iii) Isotopes of an element show same physical properties. (iv) Isotopes of an element show same chemical properties. (a)(i), (iii) and (iv) (b)(ii), (iii) and (iv) (c)(ii) and (iii) (d)(ii) and (iv). Solution: (d)(ii) and (iv)....
Read More →Which of the following elements
Question: Which of the following elements does not lose an electron easily ? (a)Na (b)F (c)Mg (d)Al. Solution: (b).The element fluorine (F) with smallest size does not lose electron easily....
Read More →Manjit bought an iron safe for ₹ 12160 and paid ₹ 340 for its transportation.
Question: Manjit bought an iron safe for ₹ 12160 and paid ₹ 340 for its transportation. Then, he sold it for ₹ 12875. Find his gain per cent Solution: CP of the iron safe = ₹12,160 Money spent on transportation = ₹340 Total CP = ₹12,160 + ₹340 = ₹12,500 SP of the iron safe = ₹12,875 Profit = SP CP = ₹12,875 ₹12,500 = ₹375 $\therefore$ Profit\% $=\frac{\text { Profit }}{\text { CP }} \times 100 \%=\frac{375}{12500} \times 100 \%=3 \%$...
Read More →Which of the following elements
Question: Which of the following elements would lose an electron easily ? (a)Mg (b)Na (c)K (d)Ca. Solution: (c).The element potassium (K) with maximum size would lose electron easily....
Read More →Solve this
Question: Find $\frac{d y}{d x}$ in each of the following: $e^{x-y}=\log \left(\frac{x}{y}\right)$ Solution: We are given with an equation $e^{x-y}=\log \left(\frac{x}{y}\right)=\log x-\log y$, we have to find $\frac{d y}{d x}$ of it, so by differentiating the equation on both sides with respect to $x$, we get, $e^{x-y}\left(1-\frac{d y}{d x}\right)=\frac{1}{x \ln 10}-\frac{1}{y \ln 10} \frac{d y}{d x}$ $\frac{d y}{d x}\left[\frac{1}{y \ln 10}-e^{x-y}\right]=\frac{1}{x \ln 10}-e^{x-y}$ $\frac{\m...
Read More →Which among the following elements
Question: Which among the following elements has the largest atomic radius ? (a)Na (b)Mg (c)K (d)Ca. Solution: (c).The element potassium (K) present in group 1 has the largest atomic radius....
Read More →Find the cost price when:
Question: Find the cost price when: (i) SP = Rs 1596 and gain $\%=12 \%$ (ii) SP = Rs 2431 and loss $\%=6 \frac{1}{2} \%$ (iii) SP = Rs $657.60$ and loss $\%=4 \%$ (iv) $\mathrm{SP}=$ Rs $34.40$ and gain $\%=7 \frac{1}{2} \%$ Solution: (i) $\mathrm{SP}=$ Rs. 1596 Gain percentage $=12 \%$ $\mathrm{CP}=\frac{100}{(100+\text { gain } \%)} \times \mathrm{SP}$ $=\frac{100}{(100+12)} \times 1596$ $=$ Rs. 1425 (ii)$\mathrm{SP}=$ Rs. 2431 Loss percentage $=6 \frac{1}{2} \%=\frac{13}{2} \%$ $\mathrm{CP}=...
Read More →Which of the following gives the correct increasing
Question: Which of the following gives the correct increasing order of the atomic radii of O, F and N ? (a)O, F, N (b)N, F, O (c)O, N, F (d)F, O, N. Solution: (d).It is the correct order. These elements are present in second period....
Read More →Find the domain and the range of each of the following real
Question: Find the domain and the range of each of the following real function: $f(x)=\frac{x-3}{2-x}$ Solution: Given: $f(x)=\frac{x-3}{2-x}$ Need to find: Where the functions are defined. Let, $f(x)=\frac{x-3}{2-x}=y$ .......(1) To find the domain of the function f(x) we need to equate the denominator of the function to 0. Therefore, $2-x=0$ $\Rightarrow x=2$ It means that the denominator is zero when $x=2$ So, the domain of the function is the set of all the real numbers except 2 . The domain...
Read More →Which one of the following elements
Question: Which one of the following elements exhibits maximum number of valence electrons ? (a)Na (b)Al (c)Si (d)P Solution: (d).The element phosphorus (P) has five electrons (2, 5) in the valence shell....
Read More →Solve this
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ in each of the following: $\tan ^{-1}\left(x^{2}+y^{2}\right)=a$ Solution: We are given with an equation $\tan ^{-1}\left(x^{2}+y^{2}\right)=a$, we have to find $\frac{d y}{d x}$ of it, so by differentiating the equation on both sides with respect to $x$, we get, $\frac{1}{x^{2}+y^{2}}\left(2 x+2 y \frac{d y}{d x}\right)=0$ $\frac{d y}{d x}=\frac{-x}{y}$...
Read More →Which of the following is the outermost
Question: Which of the following is the outermost shell for elements of period 2 ? (a)K shell (b)L shell (c)M shell (d)N shell Solution: (b).In second (2) period, the electrons are filled in second shell also known as L-shell....
Read More →An element which is an essential
Question: An element which is an essential constituent of all organic compounds belongs to (a)group 1 (b)group 14 (c)group 15 (d)group 16. Solution: (b).The element is carbon. It belongs to group 14....
Read More →Find the selling price when:
Question: Find the selling price when: (i) CP = Rs 1650 and gain $\%=4 \%$ (ii) CP $=$ Rs 915 and gain $\%=6 \frac{2}{3} \%$ (iii) $\mathrm{CP}=$ Rs 875 and loss $\%=12 \%$ (iv) CP = Rs 645 and loss $\%=13 \frac{1}{3} \%$ Solution: (i) $\mathrm{CP}=$ Rs. 1650 Gain percentage $=4 \%$ $S P=\frac{(100+\text { gain } \%)}{100} \times C P$ $=\frac{(100+4)}{100} \times 1650$ $=\frac{104 \times 1650}{100}$ $=$ Rs. 1716 (ii) CP $=$ Rs. 915 Gain percentage $=6 \frac{2}{3} \%=\frac{20}{3} \%$ $\mathrm{SP}...
Read More →Where would you locate the element
Question: Where would you locate the element with electronic configuration 2, 8 in the Modern Periodic Table ? (a)Group 8 (b)Group 2 (c)Group 18 (d)Group 10. Solution: (c).It is a noble gas element Neon (Ne) present in group 18....
Read More →Solve this
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ in each of the following: $\left(x^{2}+y^{2}\right)^{2}=x y$ Solution: $2\left(x^{2}+y^{2}\right)\left[2 x+2 y \frac{d y}{d x}\right]=y(1)+x \frac{d y}{d x}$ $\frac{d y}{d x}\left[4 y\left(x^{2}+y^{2}\right)-x\right]=y-4 x\left(x^{2}+y^{2}\right)$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}-4 \mathrm{x}\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)}{4 \mathrm{y}\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)-\mathrm{x}}$...
Read More →The elements A, B, C, D and E have atomic numbers
Question: The elements A, B, C, D and E have atomic numbers 9, 11, 17, 12 and 13 respectively. Which pair of elements belong to the same group ? (a)A and B (b)B and D (c)A and C (d)D and E. Solution: (c).Elements A (Z = 9) and C (Z = 17) belong to the same group. It is a halogen family (group 17)....
Read More →Which of the given elements A, B, C, D
Question: Which of the given elements A, B, C, D and E with atomic numbers 2, 3, 7, 10 and 30 respectively belong to the same period ? (a)A, B, C (b)B, C, D (c)A, D, E (d)B, D, E. Solution: (b).The elements B (Z = 3), C (Z = 7) and D(Z = 10) belong to the same period. It is second period....
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